Articoli correlati a Nonlinear Functional Analysis and Its Applications:...
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to the Subject.- 18 Variational Problems, the Ritz Method, and the Idea of Orthogonality.- §18.1. The Space C0?(G) and the Variational Lemma.- §18.2. Integration by Parts.- §18.3. The First Boundary Value Problem and the Ritz Method.- §18.4. The Second and Third Boundary Value Problems and the Ritz Method.- §18.5. Eigenvalue Problems and the Ritz Method.- §18.6. The Hölder Inequality and its Applications.- §18.7. The History of the Dirichlet Principle and Monotone Operators.- §18.8. The Main Theorem on Quadratic Minimum Problems.- §18.9. The Inequality of Poincaré-Friedrichs.- §18.10. The Functional Analytic Justification of the Dirichlet Principle.- §18.11. The Perpendicular Principle, the Riesz Theorem, and the Main Theorem on Linear Monotone Operators.- §18.12. The Extension Principle and the Completion Principle.- §18.13. Proper Subregions.- §18.14. The Smoothing Principle.- §18.15. The Idea of the Regularity of Generalized Solutions and the Lemma of Weyl.- §18.16. The Localization Principle.- §18.17. Convex Variational Problems, Elliptic Differential Equations, and Monotonicity.- §18.18. The General Euler-Lagrange Equations.- §18.19. The Historical Development of the 19th and 20th Problems of Hilbert and Monotone Operators.- §18.20. Sufficient Conditions for Local and Global Minima and Locally Monotone Operators.- 19 The Galerkin Method for Differential and Integral Equations, the Friedrichs Extension, and the Idea of Self-Adjointness.- §19.1. Elliptic Differential Equations and the Galerkin Method.- §19.2. Parabolic Differential Equations and the Galerkin Method.- §19.3. Hyperbolic Differential Equations and the Galerkin Method.- §19.4. Integral Equations and the Galerkin Method.- §19.5. Complete Orthonormal Systems and Abstract Fourier Series.- §19.6. Eigenvalues of Compact Symmetric Operators (Hilbert-Schmidt Theory).- §19.7. Proof of Theorem 19.B.- §19.8. Self-Adjoint Operators.- §19.9. The Friedrichs Extension of Symmetric Operators.- §19.10. Proof of Theorem 19.C.- §19.11. Application to the Poisson Equation.- §19.12. Application to the Eigenvalue Problem for the Laplace Equation.- §19.13. The Inequality of Poincaré and the Compactness Theorem of Rellich.- §19.14. Functions of Self-Adjoint Operators.- §19.15. Application to the Heat Equation.- §19.16. Application to the Wave Equation.- §19.17. Semigroups and Propagators, and Their Physical Relevance.- §19.18. Main Theorem on Abstract Linear Parabolic Equations.- §19.19. Proof of Theorem 19.D.- §19.20. Monotone Operators and the Main Theorem on Linear Nonexpansive Semigroups.- §19.21. The Main Theorem on One-Parameter Unitary Groups.- §19.22. Proof of Theorem 19.E.- §19.23. Abstract Semilinear Hyperbolic Equations.- §19.24. Application to Semilinear Wave Equations.- §19.25. The Semilinear Schrödinger Equation.- §19.26. Abstract Semilinear Parabolic Equations, Fractional Powers of Operators, and Abstract Sobolev Spaces.- §19.27. Application to Semilinear Parabolic Equations.- §19.28. Proof of Theorem 19.I.- §19.29. Five General Uniqueness Principles and Monotone Operators.- §19.30. A General Existence Principle and Linear Monotone Operators.- 20 Difference Methods and Stability.- §20.1. Consistency, Stability, and Convergence.- §20.2. Approximation of Differential Quotients.- §20.3. Application to Boundary Value Problems for Ordinary Differential Equations.- §20.4. Application to Parabolic Differential Equations.- §20.5. Application to Elliptic Differential Equations.- §20.6. The Equivalence Between Stability and Convergence.- §20.7. The Equivalence Theorem of Lax for Evolution Equations.- Linear Monotone Problems.- 21 Auxiliary Tools and the Convergence of the Galerkin Method for Linear Operator Equations.- §21.1. Generalized Derivatives.- §21.2. Sobolev Spaces.- §21.3. The Sobolev Embedding Theorems.- §21.4. Proof of the Sobolev Embedding Theorems.- §21.5. Duality in B-Spaces.- §21.6. Duality in H-Sp
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